108x^3 + 32
Understand the Problem
The question presents a polynomial expression '108x^3 + 32'. It is likely asking for simplification, factorization, or evaluation of this expression. To solve it, we would typically look for common factors or use polynomial identities.
Answer
$$ 108x^3 + 32 = 4(3x + 2)(9x^2 - 6x + 4) $$
Answer for screen readers
The factored form of the expression is: $$ 108x^3 + 32 = 4(3x + 2)(9x^2 - 6x + 4) $$
Steps to Solve
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Identify the expression The given expression is $108x^3 + 32$.
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Look for common factors First, check if there are any common factors in the coefficients 108 and 32. The greatest common divisor (GCD) of these two numbers is 4.
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Factor out the common factor Factor out 4 from the expression: $$ 108x^3 + 32 = 4(27x^3 + 8) $$
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Recognize the sum of cubes Now, we examine the expression inside the parentheses, $27x^3 + 8$. Notice that this can be expressed as a sum of cubes: $$ 27x^3 = (3x)^3 \quad \text{and} \quad 8 = 2^3 $$
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Use the sum of cubes formula Recall the sum of cubes formula: $$ a^3 + b^3 = (a + b)(a^2 - ab + b^2) $$ Setting $a = 3x$ and $b = 2$, we can apply the formula: $$ 27x^3 + 8 = (3x + 2)((3x)^2 - (3x)(2) + 2^2) $$
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Simplify the expression Now, calculate each part:
- First term: $3x + 2$
- Second term:
- $(3x)^2 = 9x^2$
- $-(3x)(2) = -6x$
- $2^2 = 4$
Putting this together gives us: $$ 27x^3 + 8 = (3x + 2)(9x^2 - 6x + 4) $$
- Combine everything Now, substitute back to the factored expression: $$ 108x^3 + 32 = 4(3x + 2)(9x^2 - 6x + 4) $$
The factored form of the expression is: $$ 108x^3 + 32 = 4(3x + 2)(9x^2 - 6x + 4) $$
More Information
Factoring polynomials is an important skill in algebra. Recognizing patterns such as the sum of cubes can simplify complex expressions significantly, making them easier to work with in further calculations.
Tips
- Ignoring common factors: Always check for common factors before proceeding to factor by patterns or identities.
- Misapplying formulas: Be careful with applying the sum of cubes formula; ensure the terms match the form $a^3 + b^3$.
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